Prove sin(x) + cos(x) /=/ 1

@Mihael_Keehl

$cosx+sinx=\sqrt{2}sin \left(x+\dfrac{\pi}{4} \right)$

$\sqrt{2}sin \left(x+\dfrac{\pi}{4} \right)=1$ only for certain values of x, certainly not for all $x \in \mathbb{R}$.

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Makes slightly more sense now, but it's a bit of a non question. It's like asking why $x^2 + 2x + 1 \neq 0$. Sum of sinusoids are sinusoids themselves, so $\sin x + \cos x$ is a sinusoidal graph and hence varies, that is, it is not a constant (straight) line.

I see yea.

Why is it sinusoidal and not a combination of them both - and also is there such thing as a cosuiod?

I see.

So just by contradiction then?

Nothing we can do by induction or anytihng else.

*counterexample

If you can provide an example where it doesn't work, then you have proven the statement is false. I'm not sure whether there is a way of solving by induction or by another method, but there would be no point in doing that if you can just do proof by counterexample.

It is a combination of them both, but you need to realise that cosine is just sine shifted backwards by a bit, so it's still a sinusoid. Cosine is a translation of sine that comes in useful so often we gave it its own name.

True but that's not really a triangle is it. But I agree stupid question unless you exclude 0 and 2pi from the domain.

Not applicable since these are trig functions but is thinking along the right lines.


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